This document contains information about positive definite matrices and eigenvectors/eigenvalues. It provides an example of a positive definite matrix with all positive eigenvalues. It also gives an example of showing that the transpose of a matrix A multiplied by itself (ATA) is positive definite if A is invertible. Finally, it provides an example of finding the eigenvectors and eigenvalues of a 2x2 matrix. The eigenvectors are solutions to Av=λv, where λ is the eigenvalue.

1. Name Sarmad Ali
Roll # 01
Class BSIT 5th
A (SEM)
Subject Linear Algebra
Institute Of SouthernPunjab (Multan)
Positive Definite Matrix=
A positive definite matrix is a symmetric matrix with all positive eigenvalues. Note that as it's a symmetric matrix all
the eigenvalues are real, so it makes sense to talk about them being positive or negative. Now, it's not always easy to tell
if a matrix ispositive definite.
Examplesof Positive Definite Matrix=
Show that if A is invertible, then ATA is positive definite.
If A isinvertible,then Av≠ 0 for any vectorv ≠ 0. Therefore vT
(AT
A)v= (vT
AT
)(Av) whichis the vectorAv dottedwithitself,thatis,the square of
the norm (orlength) of the vector.As Av ≠ 0, the normmust be positive,andtherefore vT
(AT
A)v>
Example given
2 2 -1 0
-1 2 -1
2 -1
= 3
-1 2
2 -1 0
-1 2 -1 = 8 -2 -2 = 4
0 -1 2
2,3,4> 0
0.
EigenVector and Eigen value=
If it occurs that v and w are scalar multiples, that is if. A v = λ v , {displaystyle Av=lambda v,} (1) then v is an
eigenvector of the linear transformation ,
A and the scale factor λ is the eigenvalue corresponding to thateigenvector. Equation (1) is the eigenvalue equation for
the matrix A.
Examplesof Eigen Vector and Eigen Values=
If
then the characteristic equation is

2. and the two eigenvalues are
λ1=-1, λ2=-2
All that's left is to find the two eigenvectors. Let's find the eigenvector, v1, associated with the eigenvalue, λ1=-1, first.
so clearly from the top row of the equations we get
Note that if we took the second row we would get
In either case we find that the first eigenvector is any 2 element column vector in which the two elements have equal
magnitude and opposite sign.
where k1 is an arbitrary constant. Note that we didn't have to use +1 and -1, we could have used any two quantities of
equal magnitude and opposite sign.
Going through the same procedure for the second eigenvalue: